A criterion for intersection multiplicity one
Rüdiger Achilles1985, Nagoya Mathematical Journal
Sign up for access to the world's latest research
checkGet notified about relevant papers
checkSave papers to use in your research
checkJoin the discussion with peers
checkTrack your impact
Abstract
Let X and Y be any pure dimensional subschemes of Pn k over an algebraically closed field K and let I(X) and I(Y) be the largest homogeneous ideals in K[x0,…, xn] defining X and Y, respectively. By a pure dimensional subscheme X of Pn k we shall always mean a closed pure dimensional subscheme without imbedded components, i.e., all primes belonging to I(X) have the same dimension.
Related papers
Unique irredundant intersections of completely irreducible ideals
Journal of Algebra, 2005
An ideal of a commutative ring is completely irreducible if it is not the intersection of any set of proper overideals. It is known that every ideal is an intersection of completely irreducible ideals. We characterize the rings for which every ideal can be represented uniquely as an irredundant intersection of completely irreducible ideals as precisely the rings in which every proper ideal is an irredundant intersection of powers of maximal ideals. We prove that every nonzero ideal of an integral domain R has a unique representation as an intersection of completely irreducible ideals if and only if R is an almost Dedekind domain with the property that for each proper ideal A the ring R/A has at least one finitely generated maximal ideal. We characterize the rings for which every proper ideal is an irredundant intersection of powers of prime ideals as precisely the rings R for which (i) R M is a Noetherian valuation ring for each maximal ideal M , and (ii) every ideal of R is an irredundant intersection of irreducible ideals.
The complete intersection locus of certain ideals
Journal of Pure and Applied Algebra, 1985
Linear systems and Multiplicity of ideals
The São Paulo Journal of Mathematical Sciences, 2009
The leading ideal of a complete intersection of height two, Part II
Journal of Algebra, 2007
Let (S, n) be a 2-dimensional regular local ring and let I = (f, g) be an ideal in S generated by a regular sequence f, g of length two. Let I * be the leading ideal of I in the associated graded ring gr n (S), and set R = S/I and m = n/I. In [GHK2], we prove that if µ G (I *) = n, then I * contains a homogeneous system {ξ i } 1≤i≤n of generators such that deg ξ i + 2 ≤ deg ξ i+1 for 2 ≤ i ≤ n−1, and htG(ξ1, ξ2, • • • , ξn−1) = 1, and we describe precisely the Hilbert series H(gr m (R), λ) in terms of the degrees c i of the ξ i and the integers d i , where di is the degree of Di = GCD(ξ1,. .. , ξi). To the complete intersection ideal I = (f, g)S we associate a positive integer n with 2 ≤ n ≤ c 1 + 1, an ascending sequence of positive integers (c 1 , c 2 ,. .. , c n), and a descending sequence of integers (d1 = c1, d2,. .. , dn = 0) such that ci+1 − ci > di−1 − di > 0 for each i with 2 ≤ i ≤ n − 1. We establish here that this necessary condition is also sufficient for there to exist a complete intersection ideal I = (f, g) whose leading ideal has these invariants. We give several examples to illustrate our theorems.
Some Results on Intersection Graphs of Ideals of Commutative Rings
2020
The rings considered in this article are commutative with identity which admit at least one nonzero proper ideal. Let R be a ring. Recall that the intersection graph of ideals of R, denoted by G(R), is an undirected simple graph whose vertex set is the set of all nontrivial ideals of R (an ideal I of R is said to be nontrivial if I / ∈ {(0), R}) and distinct vertices I, J are joined by an edge in G(R) if and only if I ∩ J 6= (0). Let r ∈ N. The aim of this article is to characterize rings R such that G(R) is either bipartite or 3-partite. MSC: 13A15.
Some Results on Normal Homogeneous Ideals
Communications in Algebra, 2003
In this article we investigate when a homogeneous ideal in a graded ring is normal, that is, when all positive powers of the ideal are integrally closed. We are particularly interested in homogeneous ideals in an Ngraded ring A of the form A ≥m := ℓ≥m A ℓ and monomial ideals in a polynomial ring over a field. For ideals of the form A ≥m we generalize a recent result of Faridi. We prove that a monomial ideal in a polynomial ring in n indeterminates over a field is normal if and only if the first n− 1 positive powers of the ideal are integrally closed. We then specialize to the case of ideals of the form I(λ) := J(λ), where J(λ) = (x λ1 1 ,. .. , x λn n) ⊆ K[x 1 ,. .. , x n ]. To state our main result in this setting, we let ℓ = lcm(λ 1 ,. .. , λ i ,. .. λ n), for 1 ≤ i ≤ n, and set λ ′ = (λ 1 ,. .. , λ i−1 , λ i + ℓ, λ i+1 ,. .. , λ n). We prove that if I(λ ′) is normal then I(λ) is normal and that the converse holds with a small additional assumption.
Related topics
Related papers
Intersection numbers of families of ideals
Archive for Mathematical Logic, 2013
We study the intersection number of families of tall ideals. We show that the intersection number of the class of analytic P-ideals is equal to the bounding number b, the intersection number of the class of all meager ideals is equal to h and the intersection number of the class of all F σ ideals is between h and b, consistently different from both.
A length formula for the multiplicity of distinguished components of intersections
Journal of Pure and Applied Algebra, 2001
To an arbitrary ideal I in a local ring (A; m) one can associate a multiplicity j(I; A) that generalizes the classical Hilbert-Samuel multiplicity of an m-primary ideal and which plays an important role in intersection theory. If the ideal is strongly Cohen-Macaulay in A and satisÿes a suitable Artin-Nagata condition then our main result states that j(I; M ) is given by the length of I=(x1; : : : ; x d-1 ) + x d I where d:=dim A and x1; : : : ; x d are su ciently generic elements of I . This generalizes the classical length formula for m-primary ideals in Cohen-Macaulay rings. Applying this to an hypersurface H in the a ne space we show for instance that an irreducible component C of codimension c of the singular set of H appears in the self-intersection cycle H c+1 with multiplicity e( jac H; C ; OH;C ), where jac H is the Jacobian ideal generated by the partial derivatives of a deÿning equation of H .
Multiplicities of a bigraded ring and intersection theory
Mathematische Annalen, 1997
On the regularity of products and intersections of complete intersections
2005
This paper proves that the Castelnuovo-Mumford regularities of the product and sum of two monomial complete intersection ideals are at most the sum of the regularities of the two ideals, and provides examples showing that these inequalities do not hold for general complete intersections.
Intersections of projective varieties and generic projections
Manuscripta Mathematica, 1997
Let X, Y C P~ be closed subvarieties of dimensions n and m respectively. Proving a Bezout theorem for improper intersections Stiickrad and Vogel [SVo] introduced cycles vk "= vk(X, Y) of dimension k on XNY and/~k on the ruled join variety J := J(X, Y) of X and Y which are obtained by a simple algorithm..In this paper we give an interpretation of these cycles in terms of generic projections Pk : pN ~ pn+m-k-l. For this we introduce a relative ramification locus R(Pk, X, Y) of Pk which is of dimension at most k and generalizes the usual ramification cycle in the case X = Y. We prove that this cycle is just Vk for 0 < k < dimXCIY-1. Moreover, the cycles flk+l (for -1 < k < dimXCtY-1) may be interpreted geometrically as the cycle of double points of Pk associated to the closure of the set of all (x : y) in the ruled join J such that (pk(x) : Pk(Y)) is in the diagonal A~,+~_k_: of j(pn+m-k-1, p'n+m-k-1).
Transversal intersection of monomial ideals
Proceedings - Mathematical Sciences, 2019
In this paper we prove conditions for transversal intersection of monomial ideals and derive a simplicial characterization of this phenomenon.
Characterizations of zero-dimensional complete intersections
Beiträge Zur Algebra Und Geometrie / Contributions To Algebra And Geometry, 2016
Given a 0-dimensional subscheme X of a projective space P n K over a field K , we characterize in different ways whether X is the complete intersection of n hypersurfaces. Besides a generalization of the notion of a Cayley-Bacharach scheme, these characterizations involve the Kähler and the Dedekind different of the homogeneous coordinate ring of X or its Artinian reduction. We also characterize arithmetically Gorenstein schemes in novel ways and bring in further tools such as the module of regular differential forms, the fundamental class, and the Jacobian module of X. Throughout we strive to work over an arbitrary base field K and keep the scheme X as general as possible, thereby improving several known characterizations.
Monomial Curves in Afinne Space and their Associated Prime Ideals with Six Generators as Set-Theoretic Complete Intersections
Communications - Scientific letters of the University of Zilina, 2014
It is known that k-dimensional algebraic affine variety is intersection of not fewer than n k hypersurfaces in n-dimensional affine space A. There is the presumption, that a number of these hypersurfaces is exactly n k. In this case we can say, that they are ideal-theoretic or set-theoretic complete intersections. This is also equivalent to the fact, that either the associated ideal I of this variety has generators (ideal-theoretic complete intersection) or the ideal I is radical of an ideal a, a I 3 , the ideal a has n k generators (set-theoretic complete intersection). The number n k is also height of the ideal I. The ideal is called a set-theoretic complete intersection (s.t.c.i., for short), if there are s=ht(I) elements g 1 , g 2 , g 3 ,... g s , such that rad(I)=rad(g 1 , g 2 , g 3 ,... g s ). Let K be an arbitrary field, , , , , R K x x x x 1 2 3 4 = 6 @ the polynomial ring in four variables over . , , , K C C n n n n 1 2 3 4 = ^ h a monomial curve in affine space A4 over K h...
Noncomplete intersection prime ideals in dimension $3$
Kyoto Journal of Mathematics, 2015
We describe prime ideals of height 2 minimally generated by 3 elements in a Gorenstein, Nagata local ring of Krull dimension 3 and multiplicity at most 3. This subject is related to a conjecture of Y. Shimoda and to a long-standing problem of J. Sally.
Loading Preview
Sorry, preview is currently unavailable. You can download the paper by clicking the button above.